For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

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Uniform Sampling & CDF inverse

I have a probability exam soon, and our prof told us to study the following question: "Describe a procedure for generating independent identically distributed (i.i.d.) samples of a random variable ...
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42 views

Detecting corrupted data in birthdates of a population

I have a population of N birthdates. Let's assume that birthdates are uniformly distributed over the year. I'm concerned that some of these records have been corrupted, for example by someone ...
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46 views

Cramer-Rao Uniform Distribution

If my data, $X_i\sim U(0,\theta)$ is iid. What is the Cramer-Roa lower bound for a variance estimator such as the sample variance? $ S_n= \frac{1}{n} \sum_{i=1}^n (X_i-\bar{X})^2$ I am stuck ...
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36 views

Expected value for number of occurences

Let $W$ be a random word made from letters which are in set $K$ (letters are uniformly distributed in $W$) . Suppose also that $W$ has finite length ($\geq 2$) and size of $K$ is finite ($\geq 2$). ...
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83 views

Maximum of martingales

I need to show whether or not the maximum of two martingales is also a martingale. Originally, I thought yes. But supposedly the answer is no. So as a counter-example, let $U_i$ be $iid$ $unif(0,1)$, $...
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29 views

Expectations of order statistics of uniform RVs, via exponential formulation

If $U_1,\ldots, U_n$ are i.i.d. uniform random variables, then I know that the order statistics satisfy $$(U_{(1)},\ldots, U_{(n)}) \overset{d}{=} \left(\frac{X_1}{\sum_{i=1}^{n+1} X_i}, \frac{X_1+...
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2answers
60 views

Probability of random variable with uniform distribution on an interval

Let a random variable X have a uniform distribution on the interval $[0, 10]$. Find $P(X(X + 10) > 11)$ Since X has a uniform distribution, the pdf of X is $$ f(x)=\left\{\begin{array}{ccl}c&...
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1answer
21 views

Prove that two random variables are dependent

Given two random variables X and Y where X is uniformly distributed on [-1,1] and Y = X^2, prove that these two random variables are dependent. Of course, it's clear that they are dependent. But, how ...
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75 views

$X_1$, $X_2$ i.i.d RVs, $X_1$ is uniformly distributed. Show $E\left(\frac{X_1}{X_1+X_2}\right)=\frac{1}{2}$

Let $X_1$, $X_2$ be two i.i.d. random variables and $X_1$ is uniformly distributed (discrete) on the set $\{1,2,3\}.$ Show that: $$E\left(\frac{X_1}{X_1+X_2}\right)=\frac{1}{2}$$ Can someone give me ...
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Convergence weakly to exponential random variable for this r.v.

Can I get some insight of how to solve this problem? Let $X_1, X_2, X_3, ...$ be i.i.d. copies of uniform random variable on $[0, 1]$. Let $M_n = \text{min}_{1\leq i <j \leq n} |X_i - X_j|$. Show $...
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72 views

Uniform Random Variable: Correlation and Independence

Let X be a uniform random variable defined on the interval $(0,1)$. If $Y = 6X^2−6X+1$, compute the correlation of X and Y . Are X and Y independent? Are X and Y uncorrelated? So my work is. $F(X) =...
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3answers
39 views

Find cumulative distribution function of uniform distribution

Random variable X has uniform distribution on $[0,1] \cup [2,3]$. Find cdf of variable X. I mean i do not know how to treat this on such strange interval.
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1answer
62 views

Mutual information for a continuous uniform distribution

I'm trying to compute using matlab the mutual information for an $ \infty $-PAM input (the limit of a very dense PAM constellation) for a range of snr and I got stuck. I'm working with a real-valued ...
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1answer
77 views

IID Uniform probability problem [closed]

Three students independently attempt to solve a problem. Assume that the times taken by each student to solve the problem are iid according to U(0,30). Find the probability that the student who fi ...
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22 views

Covariance of uniform random and indicator function dependant on it

Define $I = \begin{cases} 1,& \text{if } X\leq a\\ 0,& \text{if } X\gt a \end{cases}$ $X$ is uniform on $[0,1]$. We want to compute $Cov(I,X)$ which involves $E[IX]$. $E[IX] = E[IX|I=...
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22 views

Find probability from Uniform Distribution [duplicate]

Two numbers are selected at random from the interval (0,1). If these numbers are iid and uniformly distributed, find the probability that the three line segments found by breaking interval into three ...
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1answer
67 views

Collisions with four bullets

This is a follow-up question of Colliding Bullets. I'm interested in a rigorous calculation of a specific aspect of the referred question. We consider four bullets. Once per second a bullet is fired ...
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2answers
49 views

The distribution of the sum and difference of independent uniformly distributed variables

Suppose $X$ and $Y$ are independent uniformly distributed on the interval $[-a/2,a/2]$. What is the density function of $Z=X+Y$ and of $Z=X-Y$? I know that it will be the convolution of densities $...
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1answer
48 views

Uniform Probobility Distribution Word Problem, grocery store checkout and meeting a friend

The amount of time spent waiting in line at a grocery store express checkout varies from 5 minutes to 15 minutes and follows a uniform distribution. Let X be the amount of time spent waiting in line. ...
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86 views

$E[Y^2| (X-Y)^+]$ for $X,Y\stackrel{iid}\sim Unif(0,1)$

I am preparing for a test and am unsure of my workings on this exercise: Calculate $E[Y^2| (X-Y)^+]$ for $X,Y\stackrel{iid}\sim Unif(0,1)$ Where $A^+ = \max (A,0)$ My approach was to calculate ...
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61 views

Law of large number for a product of uniform iid random variables (stick breaking)

Let $(X_n)_n$, $n = 1, 2, \dots$, be an iid sequence of random variables uniformly distributed on $(0, 1]$. Set $S_0 = 1$ a.s. and, for $n = 1, 2, \dots$, set $S_n = \prod_{k=1}^n X_k$. Compare $S_n$ ...
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1answer
100 views

Probability of waiting time

Question: At a railroad junction, a car and a truck arrive between 7:15 and 7:30. A train stops the traffic for five minutes from 7:20. What is the probability that the car and truck waited for ...
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1answer
78 views

Find the pdf of $Y = g(X)$, where $X$ is a uniform random variable

The question is as follows: Let $X$ be a uniform random variable over $(-1,2)$. Let $g(x) = |x|$. Find the pdf of $Y = g(X)$. And here is my take so far: $$f(x) = \begin{cases} 1/2 & \text{ ...
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1answer
51 views

Transformation of variables for a non-monotonic function

Question: Let $U \sim \mathrm{Unif}(−α, α)$ follow the uniform distribution on the interval $(−α, α)$ for some parameter $α > 0$ and consider the transformed random variable $X = \sin(U)$. ...
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28 views

Distribution of function of uniformly random variables

I am sorry if there is no simple answer to this or the answer is completely obvious but I am approaching my wits end here. Probability isn't my forte, nor am I even a mathematician. I am essentially ...
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1answer
238 views

Uniform Distribution with Independent Random Variables to compute mean of the present value of a bond.

John wants to purchase a bond which will pay him $X$ thousand dollars after two years, where $X$ is equally likely to be any of the numbers in the set $\{0, 1, 2, 3, 4, 5\}$. John believes that the ...
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1answer
33 views

Uniform Distribution - Change of Variable

I have been stuck on the following question If $X$ has a cumulative distribution $F(x)$, then show $Y = F(X)$ has a uniform distribution with $U(0,1)$. I attempted to solve this problem by first ...
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14 views

Uniformly distributed arrival times, each willing to wait 15 minutes, what is the probability they meet? [duplicate]

Alice and Bob agree to meet for lunch on a certain day at noon. However, neither is known for punctuality. They both arrive independently at uniformly distributed times between noon and 1 pm on that ...
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3answers
41 views

density of $X^2$ when $X$ has uniform $[-1, 2]$ distribution

Suppose $X$ has uniform $[-1,2]$ distribution. I am trying to find the density of $Z=X^2$. Here is what I have done thus far: Range($Z$)$=[0,4]$. I began computing the distribution of $Z$ for $z \in ...
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1answer
37 views

When is an improper Riemann integral equal to Lebesgue integral

My original problem is given $X_i\sim^{iid}U[0,1]$, find $$\lim_{n \rightarrow \infty} (X_1X_2 \cdots X_n)^{1/n} = \lim_{n \rightarrow \infty} (\prod_{i=1}^{n} X_i)^{1/n}$$ Well, $$\lim_{n \...
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1answer
50 views

Uniform distribution on $[0,1]$ and random variable $Y=\frac{U}{e^{1-U}}$

$U$~$Unif[0,1]$ and we have the random variable $Y=\frac{U}{e^{1-U}}$. Find the density function of $Y$. So far I have that $0\le Y\le1$ and that... $$F_Y(t)=P(\frac{U}{e^{1-U}}\le t)=P(\ln (\frac{U}...
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203 views

p-value of uniformity of given distributions,Matlab

Given a vector of real numbers $[a_0,...,a_n]$, how do I find the $p$-value (in Matlab, say) that it is drawn from the uniform distribution over [0,1]? I.e. $H_0$ is the hypotheses that it is drawn ...
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2answers
74 views

Probability Xavier and Yolanda meet for lunch

Xavier and Yolanda plan to meet for lunch between noon and 1 p.m. They arrive independently with uniform distribution on [0, 1]. Yolanda will wait 30 min. for Xavier, but Xavier will only wait 15 min. ...
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1answer
34 views

$P\left(X+\frac{10}{X}>7\right)$ of a uniform distribution

Problem: $X$ has a continuous uniform distribution on $[0,10]$. Find $P\left( X + \frac{ 10 } { X } >7\right)$. So far, I have the PDF $f(x) = 1/10$ and CDF $F(x) = x/10$ for $0 < x < 10$. ...
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1answer
35 views

Density function of uniform prob distribution

Let $X ∼\operatorname{Uniform}(0,1)$. Find the density function of $Y = e^X$. I got to: $F_Y(y)$=$P(Y\le y)$=$P(e^X\le y)$=$P(X\le \ln(y))$ Not sure where to go from here?
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1answer
44 views

Uniform distribution probability calculation

Here is an exam problem with the work shown: A man and a woman agree to meet at a certain location at about 12:30 pm. The man will arrive at a time uniformly distributed between 12:15 and 12:45, ...
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Partition Theorem to show $P(W \gt Z)$

I am confused as to how to use the partition theorem on the following example? Any help is appreciated! Suppose that W has a U(0,1) distribution and suppose that W is independent of the random ...
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58 views

approximate a probability distribution by moment matching

I have a 60-40 weighted distribution, of uniform(0,7.5) and uniform(7.5,10) respectively, i.e. $$f_X(x)=(0.6/7.5)1_{x∈[0,7.5)}+(0.4/2.5)1_{x∈[7.5,1]}$$ I have worked out that $$E(X) = 0.6(7.5/2) + ...
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1answer
72 views

Variance of a weighted uniform distribution

Given a weighted uniform distribution, where it is a 60-40 mixture of uniform(0,7.5) and uniform(7.5,10), I have found the mean to be $$E(X) = 0.6(7.5/2) + 0.4((10+7.5)/2)$$ How do I find the ...
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1answer
39 views

Multivariate transformation of three independent variables

An insurance company offers the following insurance package to a customer with three businesses on the same street. The insurer will pay for all damages to the business that incurs the most damage, ...
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22 views

Identify a probability distribution with coordinate transformation

I have a problem with this task: We have a random variable $X:\Omega \rightarrow \mathbb R^2$, which is uniformly distributed on $K:= \{(x_1,x_2) \in \mathbb R^2 : \sqrt{x_1^2+x_2^2} \le 1 \}$ Now I ...
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2answers
76 views

PDF of several draws from an uniform distribution?

Suppose I draw several times from an uniform distribution, $X\sim\mathcal{U}(0, 1]$. (I'll use $\mathrm{R}()$ to denote an independent drawing.) What is then the PDF of several draws, added and/or ...
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1answer
21 views

$X$ RV with cdf $F$, $W \sim U[0,1]$ independent $\Rightarrow$ $V:=WF(X)+(1-W)F_{-}(X) \sim U[0,1]$

I try to prove: Let $X$ be a discrete random variable with cdf $F$, $F_{-}(x):=P(X<x)$, $W \sim U[0,1]$ a random variable and $X, W$ independent. Then $$V:=WF(X)+(1-W)F_{-}(X) \sim U[0,1].$$ My ...
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17 views

Mapping Function for Non-Uniform Circular Distributions

Note: I a programmer, not a mathematician. For a non-uniform distribution of points that occur on the unit circle is there a function or mapping that returns the k$th$ point in this distribution? For ...
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39 views

Uniform distribution over $\mathbb{R}^2$

Suppose, on $\mathbb{R}^2$, that $X$ is a random variable which takes values uniformly at random over the $\textit{line segment}$ from $(0,0)$ to $(a,a)$, where $a > 0$ is a positive constant. How ...
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39 views

poisson and uniform distributions

I have an answer to this question from someone else but I do not think it is right. Here is the question: Customers arrive at a bank at a Poisson rate lambda. Suppose two customers arrive during the ...
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1answer
35 views

Compound of uniform and gamma probability distributions

I am trying to compute the distribution of a uniform distribution whose upper limit is drawn from a gamma distribution. That is, $X \sim \Gamma(\alpha,\beta)$ $Y \sim U(0,X)$ We know: $f_X(x)={\...
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1answer
33 views

MLE of uniform distibution again

I've struggled for hours with a seemingly simple problem, I'm supposed to compute the MLE for $\theta$. We have $(y_1, y_2...y_n)$ obervations with a uniform distribution. The density function is as ...
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1answer
29 views

Probability of A winning given a uniform distribution

If the interval of A has been uniformly chosen as [0,1] and B as [0,6] then what is the probability of A being a lower number than B? I'm completely lost here, do I somehow calculate the uniform ...
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15 views

How do I normalize a uniform dist?

If I have a uniform distribution over A to B, and I want to find the prb of a trial being within 1 std dev, once I have the mean and std dev, how do I normalize this, so the mean is 0 and a std dev is ...